Curvature properties of exotic spheres
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== Homotopy spheres with positive [[Wikipedia:Scalar_curvature|scalar curvature]] == | == Homotopy spheres with positive [[Wikipedia:Scalar_curvature|scalar curvature]] == | ||
− | Hitchin (based on work of Lichnerowicz) proved that the so-called | + | Hitchin (based on work of Lichnerowicz) proved that the so-called $\alpha$-invariant of a closed spin manifold provides an obstruction to the existence of a metric of positive scalar curvature on it (cf. \cite{Hitchin1974}, \cite{Lichnerowicz1963}). The $\alpha$-invariant for a closed $n$-dimensional spin manifold (compare [[Spin bordism|Spin bordism Invariants]]) is given as follows: Let $Spin(M)$ the principal $Spin(n)$-bundle of $M$, and let $S$ be choice of an irreducible $\Zz/2$-graded module over the real Clifford algebra. The real spinors of $M$ are defined to be ... to be continued. |
{{beginthm|Proposition}} | {{beginthm|Proposition}} |
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1 Introduction
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A homotopy sphere of dimension is an oriented closed smooth manifold which is homotopy equivalent to the standard sphere . A homotopy sphere is called an exotic sphere if it not diffeoemorphic to a standard sphere. General information about homotopy spheres (and exotic spheres) is given on Exotic spheres. One prominent question concerning the geometry of exotic spheres is: Given an exotic sphere, are there Riemannian metrics which fulfil specific positivity criteria? One typically considers the following three types.
1 Homotopy spheres with positive sectional curvature
2 Homotopy spheres with positive Ricci curvature
3 Homotopy spheres with positive scalar curvature
Hitchin (based on work of Lichnerowicz) proved that the so-called -invariant of a closed spin manifold provides an obstruction to the existence of a metric of positive scalar curvature on it (cf. [Hitchin1974], [Lichnerowicz1963]). The -invariant for a closed -dimensional spin manifold (compare Spin bordism Invariants) is given as follows: Let the principal -bundle of , and let be choice of an irreducible -graded module over the real Clifford algebra. The real spinors of are defined to be ... to be continued.
Proposition 7.1. In case the space of harmonic spinors canonically has the structure of a complex vector space, while in case the space of harmonic spinors canonically carries the structure of a quarternionic vector space. The -invariant of a homotopy sphere is given by
2 References
- [Hitchin1974] N. Hitchin, Harmonic spinors, Advances in Math. 14 (1974), 1–55. MR0358873 (50 #11332) Zbl 0284.58016
- [Lichnerowicz1963] A. Lichnerowicz, Spineurs harmoniques, C. R. Acad. Sci. Paris 257 (1963), 7–9. MR0156292 (27 #6218) Zbl 0714.53041